Strong chaos, the butterfly effect, is a ubiquitous phenomenon in physical systems. In quantum mechanical systems, one of the diagnostics of quantum chaos is an out-of-time-order correlation function, related to the commutator of operators separated in time. In this talk we will review the work of Maldacena, Shenker and Stanford (arxiv:1503.01409), who conjectured that the influence of chaos on this correlator can develop no faster than exponentially, with Lyapunov exponent λL ≤ 2πkBT/\hbar. We will then discuss a system that displays a maximal Lyapunov exponent: the SYK model. 

Classical finite element methods discretize scalar functions using piecewise polynomials. Vector finite elements, such as those developed by Raviart-Thomas, Nédélec, and Brezzi-Douglas-Marini in the 1970s and 1980s, have since undergone significant theoretical advancements and found wide-ranging applications. Subsequently, Bossavit recognized that these finite element spaces are specific instances of Whitney’s discrete differential forms, which inspired the systematic development of Finite Element Exterior Calculus (FEEC). These discrete topological structures and patterns also emerge in fields like Topological Data Analysis.

In this talk, we present an overview of discrete and finite element differential forms motivated by applications from topological hydrodynamics, alongside recent advancements in tensorial finite elements. The Bernstein-Gelfand-Gelfand (BGG) sequences encode the algebraic and differential structures of tensorial problems, such as those encountered in solid mechanics, differential geometry, and general relativity. Discretization of the BGG sequences extends the periodic table of finite elements, originally developed for Whitney forms, to include Christiansen’s finite element interpretation of Regge calculus and various distributional finite elements for fluids and solids as special cases. This approach further illuminates connections between algebraic and geometric structures, generalized continuum models, finite elements, and discrete differential geometry.

TBA